Difference between revisions of "2023 AMC 10B Problems/Problem 17"
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Incorporating the solution above, the side lengths are larger than <math>1x1x1</math> (a unit cube). The side length of the interior of a unit cube is <math>\sqrt{3}</math>, and we know that the side lengths are larger than <math>1x1x1</math>, so that means the diagonal has to be larger than <math>\sqrt{3}</math>, and there is only one answer choice larger than <math>\sqrt{3}</math> <math>\Rightarrow</math> <math>\boxed {\textbf{(D) 9/4}}</math> ~ kabbybear | Incorporating the solution above, the side lengths are larger than <math>1x1x1</math> (a unit cube). The side length of the interior of a unit cube is <math>\sqrt{3}</math>, and we know that the side lengths are larger than <math>1x1x1</math>, so that means the diagonal has to be larger than <math>\sqrt{3}</math>, and there is only one answer choice larger than <math>\sqrt{3}</math> <math>\Rightarrow</math> <math>\boxed {\textbf{(D) 9/4}}</math> ~ kabbybear |
Revision as of 18:22, 15 November 2023
Contents
Problem
A rectangular box 𝒫 has distinct edge lengths 𝑎, 𝑏, and 𝑐. The sum of the lengths of
all 12 edges of 𝒫 is 13, the sum of the areas of all 6 faces of 𝒫 is , and the volume of 𝒫 is
. What is the length of the longest interior diagonal connecting two vertices of 𝒫 ?
Solution 1
Let
and
be the sides of the box, we get
The diagonal of the box is
~Technodoggo
Note
Interestingly, we don't use the fact that the volume is ~andliu766
Solution 2 (find side lengths)
Let be the edge lengths.
Then, you can notice that these look like results of Vieta's formula:
Finding when this
will give us the edge lengths.
We can use RRT to find one of the roots:
One is
, dividing gives
.
The other 2 roots are
Then, once we find the 3 edges being and
, we can plug in to the distance formula to get
.
-HIA2020
Solution 3 (Cheese method)
Incorporating the solution above, the side lengths are larger than (a unit cube). The side length of the interior of a unit cube is
, and we know that the side lengths are larger than
, so that means the diagonal has to be larger than
, and there is only one answer choice larger than
~ kabbybear